First slide

Trigonometric equations

Question

For $x \in (0, π)$ , the equation $\sin x + 2 \sin 2 x - \sin 3 x = 3$ has

Difficult

A

Infinitely many solutions
B

3 solutions
C

one solution
D

No solution

Solution

$\sin x + 2 \sin 2 x - \sin 3 x = 3$ $\Rightarrow \sin x - 4 \sin x \cos x - 3 \sin x + 4 \sin^{3} x = 3$
$\Rightarrow - 2 - 4 \cos x + 4 \sin^{2} x = 3 \cos e c x (d i v i d i n g t h r o u g h o u t b y \sin x)$
$\Rightarrow 2 - 4 \cos^{2} x - 4 \cos x = 3 \cos e c x$
$\Rightarrow 3 - 4 {(\cos x - \frac{1}{2})}^{2} = 3 \cos e c x$
L.H.S $< 3$ ; R.H.S $\geq 3$ , No solution.

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